Yes, you will have issues on PHP 5.4 version, composer update will not work. General recommendation is upgrade PHP to newest version. Only exception to this rule is legacy nightmare apps.
If your hosting company does not support at least 5.6 (and it really should be 7+) it is not a company worth using.
That said I would be surprised if a hosting company as large as Godaddy doesn't offer newer versions of php. You might need to contact their support to be upgraded.